It's important to use an oil which doesn't creep away from the pivots. The Hoppe's oil will creep. A very light molybdenum grease is good for points where (for example) a spring end bears and rubs on some other part.
Lubricants have improved drastically since these old shutters were made, I have two old Compounds from before the War To End All War, (WWI) specifically 1908 and 1913 - after overhaul they are as accurate as new modern shutters.
Edit - I have no connection (beyond being a happy customer) with the supply house I posted the link to.
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